// poly43.txt  fourth order polinomial in three dimensions
// integer coefficients used for convenience and counting

U(x,y,z) = 1.0*x*x*x*x +  2.0*x*x*x*y +  3.0*x*x*x*z +  4.0*x*x*y*y +
           5.0*x*x*y*z +  6.0*x*x*z*z +  7.0*x*y*y*y +  8.0*x*y*y*z +
           9.0*x*y*z*z + 10.0*x*z*z*z + 11.0*y*y*y*y + 12.0*y*y*y*z +
          13.0*y*y*z*z + 14.0*y*z*z*z + 15.0*z*z*z*z +

          16.0*x*x*x   + 17.0*x*x*y   + 18.0*x*x*z   + 19.0*x*y*y   +
          20.0*x*y*z   + 21.0*x*z*z   + 22.0*y*y*y   + 23.0*y*y*z   +
          24.0*y*z*z   + 25.0*z*z*z   +

          26.0*x*x     + 27.0*x*y     + 28.0*x*z     + 29.0*y*y     +
          30.0*y*z     + 31.0*z*z     +

          32.0*x       +33.0*y        + 34.0*z       + 35.0    

There are also exactly 15 derivatives of x, y, z at 4th order
each with the value of the corfficient above

dU(x,y,z)/dxxxx =  1.0*4*3*2
dU(x,y,z)/dxxxy =  2.0*3*2
dU(x,y,z)/dxxxz =  3.0*3*2
dU(x,y,z)/dxxyy =  4.0*2*2
dU(x,y,z)/dxxyz =  5.0*2
dU(x,y,z)/dxxzz =  6.0*2*2
dU(x,y,z)/dxyyy =  7.0*3*2
dU(x,y,z)/dxyyz =  8.0*2
dU(x,y,z)/dxyzz =  9.0*2
dU(x,y,z)/dxzzz = 10.3*2
dU(x,y,z)/dyyyy = 11.4*3*2
dU(x,y,z)/dyyyz = 12.0*3*2
dU(x,y,z)/dyyzz = 13.0*2*2
dU(x,y,z)/dyzzz = 14.0*3*2
dU(x,y,z)/dzzzz = 15.0*4*3*2

The derivatives at 3rd order are

dU(x,y,z)/dxxx =  16.0 + 1.0*4*3*2*x + 2.0*3*2 + 3.0*3*2
dU(x,y,z)/dxxy =  17.0 +
dU(x,y,z)/dxxz =  18.0 +
dU(x,y,z)/dxyy =  19.0 +
dU(x,y,z)/dxyz =  20.0 +
dU(x,y,z)/dxzz =  21.0 +
dU(x,y,z)/dyyy =  22.0 +
dU(x,y,z)/dyyz =  23.0 +
dU(x,y,z)/dyzz =  24.0 +
dU(x,y,z)/dzzz =  25.0 +

The derivatives at 2nd order are

dU(x,y,z)/dxx =  26.0 + 1.0*4*3*x*x + 2.0*3*2*y + 3.0*3*2*z +
dU(x,y,z)/dxy =  27.0 +
dU(x,y,z)/dxz =  28.0 +
dU(x,y,z)/dyy =  29.0 +
dU(x,y,z)/dyz =  30.0 +
dU(x,y,z)/dzz =  31.0 +

The derivatives at 1st order are

dU(x,y,z)/dx =  32.0 + 1.0*4*x*x*x + 2.0*3*x*x*y + 3.0*3*x*x*z +
dU(x,y,z)/dy =  33.0 + 2.0*x*x*x + 
dU(x,y,z)/dz =  34.0 + 3.0*x*x*x +